In recent years, the Natural Language Processing community have been moving from uncontextualized word embeddings towards contextualized word embeddings. Among these contextualized architectures, BERT stands out due to its capacity to compute bidirectional contextualized word representations. However, its competitive performance in English downstream tasks is not obtained by its multilingual version when it is applied to other languages and domains. This is especially true in the case of the Spanish language used in Twitter.In this work, we propose TWiLBERT, a specialization of BERT architecture both for the Spanish language and the Twitter domain. Furthermore, we propose a Reply Order Prediction signal to learn inter-sentence coherence in Twitter conversations, which improves the performance of TWilBERT in text classification tasks that require reasoning on sequences of tweets. We perform an extensive evaluation of TWilBERT models on 14 different text classification tasks, such as irony detection, sentiment analysis, or emotion detection. The results obtained by TWilBERT outperform the state-of-the-art systems and Multilingual BERT. In addition, we carry out a thorough analysis of the TWilBERT models to study the reasons of their competitive behavior. We release the pre-trained TWilBERT models used in this paper, along with a framework for training, evaluating, and fine-tuning TWilBERT models.